Simplify the following expression and state the condition under which the simplification is valid: $t = \dfrac{z^2 - 10z}{z^2 - 19z + 90}$
Explanation: First factor the expressions in the numerator and denominator. $ \dfrac{z^2 - 10z}{z^2 - 19z + 90} = \dfrac{(z)(z - 10)}{(z - 9)(z - 10)} $ Notice that the term $(z - 10)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(z - 10)$ gives: $t = \dfrac{z}{z - 9}$ Since we divided by $(z - 10)$, $z \neq 10$. $t = \dfrac{z}{z - 9}; \space z \neq 10$